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Charged Rotating Black Hole and the First Law

S. D Campos

TL;DR

This work extends black hole thermodynamics to charged rotating BHs by leveraging a charged rotating soap-bubble analogy and the Gouy-Stodola theorem to obtain a consistent first-law framework. It shows that the standard Kerr–Newman relation $dM = \frac{\kappa}{8\pi} dA + \Omega dJ + \Phi dQ$ remains valid, while charge influences entropy production through electromagnetic angular momentum. The partition function analysis reveals a distance-dependent behavior where distant observers perceive a nearly constant energy state, yet near-horizon physics retains charge effects, with potential implications for Hawking radiation and BH evaporation. Overall, the approach strengthens the thermodynamic interpretation of BHs and clarifies how charge, rotation, and entropy interplay in gravitational systems, potentially guiding observational probes in astrophysical and gravitational-wave data.

Abstract

The thermodynamic properties of black holes have been extensively studied through analogies with classical systems, revealing fundamental connections between gravitation, entropy, and quantum mechanics. In this work, we extend the thermodynamic framework of black holes by incorporating charge and analyzing its role in entropy production. Using an analogy with charged rotating soap bubbles, we demonstrate that charge contributes to the total angular momentum and affects the entropy-event horizon relationship. By applying the Gouy-Stodola theorem, we establish a consistent thermodynamic formulation for charged black holes, showing that the first law of thermodynamics remains valid in this context. Furthermore, we explore the behavior of the partition function from the perspective of a distant observer, revealing that charge effects diminish with increasing distance. These findings reinforce the thermodynamic interpretation of black holes and provide insights into the interplay between charge, rotation, and entropy in gravitational systems.

Charged Rotating Black Hole and the First Law

TL;DR

This work extends black hole thermodynamics to charged rotating BHs by leveraging a charged rotating soap-bubble analogy and the Gouy-Stodola theorem to obtain a consistent first-law framework. It shows that the standard Kerr–Newman relation remains valid, while charge influences entropy production through electromagnetic angular momentum. The partition function analysis reveals a distance-dependent behavior where distant observers perceive a nearly constant energy state, yet near-horizon physics retains charge effects, with potential implications for Hawking radiation and BH evaporation. Overall, the approach strengthens the thermodynamic interpretation of BHs and clarifies how charge, rotation, and entropy interplay in gravitational systems, potentially guiding observational probes in astrophysical and gravitational-wave data.

Abstract

The thermodynamic properties of black holes have been extensively studied through analogies with classical systems, revealing fundamental connections between gravitation, entropy, and quantum mechanics. In this work, we extend the thermodynamic framework of black holes by incorporating charge and analyzing its role in entropy production. Using an analogy with charged rotating soap bubbles, we demonstrate that charge contributes to the total angular momentum and affects the entropy-event horizon relationship. By applying the Gouy-Stodola theorem, we establish a consistent thermodynamic formulation for charged black holes, showing that the first law of thermodynamics remains valid in this context. Furthermore, we explore the behavior of the partition function from the perspective of a distant observer, revealing that charge effects diminish with increasing distance. These findings reinforce the thermodynamic interpretation of black holes and provide insights into the interplay between charge, rotation, and entropy in gravitational systems.

Paper Structure

This paper contains 7 sections, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Charged Rotating
  3. Soap Bubbles
  4. Black Holes
  5. First Law for a Charged Rotating Black Hole
  6. Partition Function
  7. Discussion

Figures (3)

  • Figure 1: Charge stored by the electromagnetic angular momentum for a spherical soap bubble, where one uses $r=0.112$ m, and $\sigma=34$ mF/m from Ref. A.Pelesz.P.Zylka.Exp.in.Fluids.61.241.2020. Observe that the radius $r=0.6$ m corresponds to a huge soap bubble.
  • Figure 2: Charge stored by the electromagnetic momentum angular for the BH case for $M=10$ MeV and $\delta J=0.5$. It should be noted that, analogous to the case of the spherical soap bubble, the charge diminishes with the increase of $b$.
  • Figure 3: Charge variation according to $\theta$ for $J=0.1$, $R=1$ MeV$^{-1}$ and $M=10$ MeV, and respecting the constraint \ref{['eq:constraint']}.